Analysis of Eclipsing Binary Data

analysis of eclipsing binary data A.E. Lynas-Gray and C. Causer October 2012 AS33 1 Introduction The purpose of this experiment is to verify physical and geometric characteristics of an eclipsing binary already published in the literature, using radial velocity and light curves provided. In carrying out the experiment, the most important thing is to appreciate how the masses, radii, shapes, temperatures and orbits of the two stars determine the observed light and radial velocity variations. The book by Hilditch (2001) is an excellent introduction to binary stars and is recommended reading. Once the period has been determined, masses, radii and effective temperatures of both components are estimated using a “back of an envelope” calculation; these results are then used as a starting approximation in a light and radial velocity curve synthesis code. The idea is to reproduce observed light and radial velocity curves, but in practice “back of the envelope” results will need adjustment before a good agreement with observation is achieved. At the time of writing, only data for R Canis Majoris are provided. 2 Initial setup Begin by creating a xterm window in which commands can be typed. The window is created using the Mouse to click on the large “X” in the middle of the dock on the bottom of your screen. The large “X” may bob up and down a bit before the window appears. The steps detailed below provide the setup necessary for experiments carried out in the Astrophysics Laboratory. The .profile, .bashrc, .cups/lpoptions and .Xdefaults files in your home directory are changed as a result of carrying out steps listed below; if these files have been previously setup to carry out an experiment in another laboratory, please consult a demonstrator before proceeding. If you have already carried an experiment in the Astrophysics Laboratory and the .profile, .bashrc, .cups/lpoptions and .Xdefaults files in your home directory have remained unchanged since, skip the steps below and move on to the next section. Provide yourself with copies of the necessary .profile, .bashrc, .cups/lpoptions and .Xdefaults files by executing the commands $ cp ~aelg/astro-lab/profile .profile $ cp ~aelg/astro-lab/bashrc .bashrc 03Oct2012 Copyright c 2012 University of Oxford, except where indicated. AS33- 1 Analysis of eclipsing binary data $ cp ~aelg/astro-lab/Xdefaults .Xdefaults $ cp ~aelg/astro-lab/lpoptions .cups/lpoptions where the leading ”$” represents the operating system command line prompt which by default is some long string. Once the above commands have been typed, logout and login again; this causes the .profile, .bashrc, .cups/lpoptions and .Xdefaults files to be executed and so providing the setup necessary to conduct experiments in the Astrophysics Laboratory. Create a new xterm window, as described above, in which commands can be typed; the first obvious sign of the new setup is the replacement of the default operating system command line prompt with a simple “$”. It is worth noting some unorthodox inputting methods inherent to Mac keyboards and mice. Some computers in the Lab have mice with a right click feature (which is turned on in “System Preferences.”), For mice without this feature (the clear ones), a right click can be emulated by using ctrl + left click. Also, programs such as NIGHTFALL use the # symbol in their data files; this is typed by pressing shift + 3 (the £ symbol is typed by pressing alt + 3). 3 Getting started All computer commands reproduced above and below for your guidance are given in typewriter font. The leading "$ " represents the command line prompt. A leading “> ” represents a prompt provided by an executing program such as dipso. Typewriter font is also used to specify file names, file contents and environment variables. Further necessary information is provided in the Third Year Astrophysics Laboratory Practicals page http://www-astro.physics.ox.ac.uk/~aelg/Third_Year_Laboratory which can be read with a browser such as safari. The first step is to create a new directory in which all files, associated with the experiment described here, will be stored. Issue a command of the form $ mkdir as33 to create such a directory. Then type $ cd as33 to make the new directory your current working directory. Data files will need to be edited as described below, or a decision (justified in the material submitted for marking) may be taken to discard one or more observations. Write access to the data files is also required by the programs used although the files are not changed by them. Copies of the light curve and radial velocity files are therefore made in the working directory. The commands issued depend on the star being analysed; in the case of R Canis Majoris $ cp ~aelg/eclipsing_binary/data/R_CMA/B.dat . AS33-2 Copyright c 2012 University of Oxford, except where indicated. Analysis of eclipsing binary data $ cp ~aelg/eclipsing_binary/data/R_CMA/V.dat . $ cp ~aelg/eclipsing_binary/data/R_CMA/rv1.dat . $ cp ~aelg/eclipsing_binary/data/R_CMA/rv2.dat . are typed; the full stop at the end specifies that the file is to have the same name when copied to the current directory. Each file consists of three columns which from left to right give time (as Heliocentric Julian date expressed in days), the observation (in magnitudes or kilometres per second) and the standard deviation in the observation (in the same units as used for the observation). The files B.dat and V.dat contain respectively Johnson B-band and V-band differential photometry; rv1.dat and rv2.dat contain radial velocities for the more massive and less massive star respectively. In the case of R Canis Majoris, differential photometry is taken from Radhakrishnan & Sarma (1982) and radial velocities from Tomkin (1985). Radhakrishnan & Sarma use BD −15o1732 as their comparison star and for which they obtain V = 5.48 ± 0.01, (B − V) = 0.07 ± 0.01 and (U − B) = 0.04 ± 0.01. The epoch (see below) adopted by Radhakrishnan & Sarma, and to be used here for phasing all observations, is 2444648.3283. Ribas et al. (2002) give a good overview R Canis Majoris, along with references to earlier work, 4 Period determination Eclipsing binaries generally have light curves which exhibit well defined Primary Minima and the time intervals between these events can be a good indicator of the period; it is never an exact indicator, of course, because it is unlikely that an observation was ever made at the exact time of Primary Minimum. Write down the approximate times of Primary Minima as seen in the Johnson B-band and Johnson V-band light curves and look at the intervals between them. The smallest interval between any two Primary Minima is either the period or some integral multiple of it. Try to improve the estimate of the period by taking the average of the smallest intervals. Are the larger intervals exact multiples of the smaller intervals? If not, use a larger interval to obtain a more precise estimate of the period. How can the possibility that the derived period is a multiple of the true period be excluded? Once a period has been determined, phases for each observation can be calculated using a relation of the form f = (T − T0)/P (1) where P is the period in days, T is the Heliocentric Julian date of observation (also in days) and T0 is the Heliocentric Julian date of some reference observation (also in days) at which the phase is defined to be zero. In the case of an eclipsing binary, it makes sense to select T0 as a time of Primary Minimum so that when the correct period is used all Primary Minima are at f = 0 or integral values of f. Because the times of Primary Minima are not measured precisely, the period derived using them can at best be a starting approximation for an improved technique which uses all observations. The improved technique adopted here is the “string-length” method which Dworetsky (1983) proposes. Once the correct period is used in (1), the plot of observation (magnitude or radial velocity) as a function of phase results in a smooth curve; that is the the length of the piecewise line joining each point (or “string- length”) is minimised. The program strphase plots observations as a function of phase for a range of trial periods; in each case it also gives the “string-length” (l) calculated using n−1 1/2 1/2 h 2 2i h 2 2i l = ∑ (oi − oi−1) + (fi − fi−1) + (o1 − on) + (1 + f1 − fn) (2) i=2 Copyright c 2012 University of Oxford, except where indicated. AS33-3 Analysis of eclipsing binary data th where oi is the i observation scaled as Dworetsky (1983) recommends. In calculating the standard deviation (s(l)) in the “string-length” using strphase, it is assumed that each trial phase is known exactly. Moreover, each observation is independent and so uncorrelated with any other; we therefore have n 2 2 ¶l 2 s (l) = 2 ∑ s (oi) (3) i=1 ¶oi th 2 th where the variance in the i scaled observation (s (oi)) is related to the variance in the unscaled i 2 observation (s (Oi)) by 2 2 2 2 ¶oi 2 ¶oi 2 ¶oi 2 s (oi) = s (Oi) + s (Omax) + s (Omin) (4) ¶Oi ¶Omax ¶Omin with ¶o 1 i = , (5) ¶Oi 2(Omax − Omin) ¶o O − O i = i max 2 , (6) ¶Omin 2(Omax − Omin) and ¶o O − O i = − i min 2 . (7) ¶Omax 2(Omax − Omin) Here Omax and Omin are respectively the maximum and minimum unscaled observations. Before running strphase it is necessary to tell it which set of observations to use; this is done through setting an environment variable.

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